mathpazo old-style figures and eulervm: I want lining figures in math

You have to use "lining figures" fonts for the math alphabets: add the following lines before \begin{document}:

\DeclareSymbolFont{operators}{\encodingdefault}{ppl}{m}{n}
\DeclareMathAlphabet{\mathbf}{\encodingdefault}{ppl}{bx}{n}
\DeclareMathAlphabet{\mathit}{\encodingdefault}{ppl}{m}{it}

so that the alphabets chosen by \mathrm, \mathbf and \mathit will use the family ppl rather than the default family.

Example

\documentclass{article}

\usepackage[utf8]{inputenc}
\usepackage[sc,osf]{mathpazo}
\usepackage[euler-digits]{eulervm}
\usepackage[T1]{fontenc}
\usepackage{mathtools}

\DeclareSymbolFont{operators}{\encodingdefault}{ppl}{m}{n}
\DeclareMathAlphabet{\mathbf}{\encodingdefault}{ppl}{bx}{n}
\DeclareMathAlphabet{\mathit}{\encodingdefault}{ppl}{m}{it}

\begin{document}

Mathpazo text: 1234567890 old-style figures OK.

Euler math with euler-digits:
\begin{align*}
1234567890  \qquad &\text{mathnormal: euler-digits OK}  \\
\mathrm{1234567890}  \qquad &\text{mathrm: pazo old-style figures OK}  \\
\mathit{1234567890}  \qquad &\text{mathit: pazo old-style figures italic OK}  \\
\mathbf{1234567890}  \qquad &\text{mathbf: pazo old-style figures bold OK}  \\
\mathbold{1234567890}  \qquad &\text{mathbold: euler-digits OK}
\end{align*}$

\end{document}